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The resolution of excess investment in Fuzzy portfolio selection using piecewise linear function

APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 2024年第1期32卷

作者： Huang, Yin-Yin Ma, Tongxia Pan, Jingchao Tsaur, Ruey-Chyn Nanchang Vocat Univ Sch Econ & Management Nanchang Peoples R China GongQing Inst Sci & Technol Jiujiang Peoples R China Tamkang Univ Dept Management Sci New Taipei City 25137 Taiwan

Fuzzy portfolio selection on the guaranteed return rate has been proposed to study excess investment in different dimensions of risk preferences. However, the theory of behavioural finance proposes that the investment proportions of securities are not only determined by the intrinsic value of the security but also largely affected by the behaviour of investors. Therefore, the investment behaviour of an investor in the fuzzy portfolio selection should be considered in the excess investment for different securities at different return rates. In this study, we propose a piecewise linear function for the fuzzy expected return rates and use the new decision variables for investment proportion in each security. In addition, considering the investment behaviours of decision-makers, we use the concept of fuzzy contingent strategy to take into account the relationships of investment proportion among the securities in the fuzzy portfolio selection. Analysis results indicated that we can find the investment proportions in the portfolio with the possible results of shortage investment, full investment, or excess investments by the fuzzy constraint of total investment proportions. Finally, in the illustration with two experiments, investors could individually select the portfolio by the piecewise linear function in shortage investment, full investment, or excess investment.

关键词： Fuzzy portfolio selection excess investment guaranteed return rate piecewise linear function contingent strategy 90-10

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A Comparison of Two Mixed-Integer linear Programs for piecewise linear function Fitting

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INFORMS JOURNAL ON COMPUTING 2022年第2期34卷 1042-1047页

作者： Warwicker, John Alasdair Rebennack, Steffen Karlsruhe Inst Technol Inst Operat Res Stochast Optimizat D-76185 Karlsruhe Raden Wurttembe Germany

The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507-530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531-5461 Both formulations are similar in that they use binary variables and logical implications modelled by big-M constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two

关键词： piecewise linear function mixed-integer linear programming function fitting

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Smoothed piecewise linear Lyapunov function for the First Order Dynamical Systems

Smoothed Piecewise Linear Lyapunov Function for the First Or...

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International Scientific and Technical Conference on Integrated Computer Technologies in Mechanical Engineering - Synergetic Engineering (ICTM)

作者： Voliansky, Roman Shramko, Iurii Volianska, Nina Tolochko, Olga Sadovoi, Oleksandr Natl Tech Univ Ukraine Igor Sikorsky Kyiv Polytech Inst 37 Str Polytech UA-03056 Kiev Ukraine Dniprovsk State Tech Univ 2 Dniprobudivska Str UA-51918 Kamyanske Ukraine

ISBN: (纸本)9783031614149;9783031614156

The paper deals with the mathematical backgrounds of design piecewise single-variable Lyapunov functions. These backgrounds are based on the study of the main features of generalized Lyapunov function and using them to analyze the solution of dynamical programming problem. This analysis shows that the optimal control signal, which is supplied to the control plant, depends on the partial derivative of the Lyapunov function. This fact allows us to consider the Lyapunov function as a control signal integral over the perturbed motion variable. If the control signal is produced by the sliding mode controller, our approach defines the Lyapunov function as a non-quadratic absolute value function. Since this function is not differentiable in the origin, it is hard enough to use it while the optimal controller is being designed with optimal theory methods. We avoid this problem by applying a smoothing procedure, which is based on considering a small neighborhood near the fracture point. In this neighborhood, we replace the piecewise linear function with a polynomial one. Factors of the polynomial are defined to have the same function values and its derivatives in the boundary points. In our paper, we show that our approach can be extended to the case of a controller which produces a multilevel signal. Signal of such form causes occurring the piecewise linear Lyapunov function, which can be smoothed by using polynomials. Since such a piecewise function can have a lot of branches, we offer to consider it as two functions, which are defined in their intervals.

关键词： Lyapunov function piecewise linear function sliding mode control polynomial smoothing

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piecewise linear function Fitting via Mixed-Integer linear Programming

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INFORMS JOURNAL ON COMPUTING 2020年第2期32卷 507-530页

作者： Rebennack, Steffen Krasko, Vitally Karlsruhe Inst Technol Inst Operat Res D-76185 Karlsruhe Baden Wurttembe Germany Colorado Sch Mines Div Econ & Business Golden CO 80401 USA

piecewise linear (PWL) functions are used in a variety of applications. Computing such continuous PWL functions, however, is a challenging task. Software packages and the literature on PWL function fitting are dominated by heuristic methods. This is true for both fitting discrete data points and continuous univariate functions. The only exact methods rely on nonconvex model formulations. Exact methods compute continuous PWL function for a fixed number of breakpoints minimizing some distance function between the original function and the PWL function. An optimal PWL function can only be computed if the breakpoints are allowed to be placed freely and are not fixed to a set of candidate breakpoints. In this paper, we propose the first convex model for optimal continuous univariate PWL function fitting. Dependent on the metrics chosen, the resulting formulations are either mixed-integer linear programming or mixed-integer quadratic programming problems. These models yield optimal continuous PWL functions for a set of discrete data. On the basis of these convex formulations, we further develop an exact algorithm to fit continuous univariate functions. Computational results for benchmark instances from the literature demonstrate the superiority of the proposed convex models compared with state-of-the-art nonconvex models.

关键词： piecewise linear function linear spline splines of degree 1 polyhedral function mixed-integer linear programming (MILP) mixed-integer quadratic programming (MIQP) function fitting spline regression global optimization

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Matching localization algorithm of nonlinear T-S fuzzy system constructed by the piecewise linear function

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JOURNAL OF AMBIENT INTELLIGENCE AND HUMANIZED COMPUTING 2019年第2期10卷 417-427页

作者： Wang, Guijun Zhang, Guoying Tianjin Normal Univ Sch Math Sci Tianjin 300387 Peoples R China

The piecewise linear function is a generalization of a segmentation linear functions with one variable in the multivariate case, which is an important tool to describe the relationships between fuzzy system and approximation function. Firstly, aiming at the -integrable function, the concrete method of constructing piecewise linear functions and their analytical expressions were introduced, furthermore, based on the piecewise linear functions, we constructed the nonlinear T-S fuzzy systems with a singleton fuzzifier in this paper. And then, we put forward the matching-localization algorithm of the fuzzy system based on the spatial positioning mode for the input variables in the domain. Finally, the effectiveness of the matching localization algorithm of the nonlinear T-S fuzzy system was verified by the means of the simulation example. The results show that the approximating ability of the fuzzy system to a <-integrable function can be adjusted by changing the subdivision parameters.

关键词： piecewise linear function Singleton fuzzifier Nonlinear T-S fuzzy system (mu)over-cap-integrable function Matching-localization algorithm

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Modeling the memristor with piecewise linear function

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INTERNATIONAL JOURNAL OF NUMERICAL MODELLING-ELECTRONIC NETWORKS DEVICES AND FIELDS 2015年第1期28卷 96-106页

作者： Mu, Xiaomu Yu, Juntang Wang, Shuning Tsinghua Univ Tsinghua Natl Lab Informat Sci & Technol Beijing 100084 Peoples R China Tsinghua Univ Dept Automat Beijing 100084 Peoples R China

Memristor is introduced as the fourth basic circuit element. In this paper, we show that piecewise linear function can be used to model memristor. piecewise linear function is used as a window function to model the memristor device. The piecewise linear window function is linear at each subregion of the domain. Because of this good character, explicit relation between the thickness of the doped region and the charge can be delivered, together with the explicit memristance expression. Models using a general nonlinear window function may not have these explicit results. The piecewise linear window function is also more flexible than the existing window functions. piecewise linear function can also be used to model the more general memristive systems. Copyright (c) 2014 John Wiley & Sons, Ltd.

关键词： memristor piecewise linear function window function memristive system

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Representing piecewise linear functions by functions with small arity

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2025年第4期36卷 595-610页

作者： Koutschan, Christoph Moser, Bernhard Ponomarchuk, Anton Schicho, Josef Johann Radon Inst Computat & Appl Math RICAM Linz Austria Software Competence Ctr Hagenberg SCCH Hagenberg Austria Johannes Kepler Univ JKU Res Inst Symbol Computat RISC Linz Austria

A piecewise linear function can be described in different forms: as a nested expression of min - and max-functions, as a difference of two convex piecewise linear functions, or as a linear combination of maxima of affine-linear functions. In this paper, we provide two main results: first, we show that for every piecewise linear function f : R-n -> R, there exists a linear combination of max-functions with at most n + 1 arguments, and give an algorithm for its computation. Moreover, these arguments are contained in the finite set of affine-linear functions that coincide with the given function in some open set. Second, we prove that the piecewise linear function max(0, x(1),., x(n)) cannot be represented as a linear combination of maxima of less than n + 1 affine-linear arguments. This was conjectured by Wang and Sun (IEEE Trans Inf Theory 51:4425-4431, 2005) in a paper on representations of piecewise linear functions as linear combination of maxima.

关键词： piecewise linear function linear combination of max-functions ReLU network Convex polyhedra Minkowski sum Zero volume polytope

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Approximation factor of the piecewise linear functions in Mamdani fuzzy system and its realization process

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JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2021年第6期41卷 6859-6873页

作者： Tao, Yujie Suo, Chunfeng Wang, Guijun Tonghua Normal Univ Sch Math Tonghua Jilin Peoples R China Beihua Univ Sch Math & Stat Jilin Jilin Peoples R China Tianjin Normal Univ Sch Math Sci Tianjin Peoples R China

piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.

关键词： piecewise linear function mesh subdivision approximation factor Mamdani fuzzy system matrix norm

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MASL-AFU: A High Memory Access Efficiency 2-D Scalable LUT-Based Activation function Unit for On-Device DNN Training

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IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS 2025年第3期33卷 707-719页

作者： Meng, Zhaoteng Shu, Lin Zeng, Jianing Li, Zhan Lv, Kailin Yang, Haoyue Hao, Jie Chinese Acad Sci Inst Automat Beijing 100045 Peoples R China Univ Chinese Acad Sci Sch Artificial Intelligence Beijing 101408 Peoples R China Guangdong Inst Artificial Intelligence & Adv Comp Guangdong 510535 Peoples R China

On-device deep neural network (DNN) training faces constraints in storage capacity and energy supply. Existing works primarily focus on optimizing the training of convolutional and batch normalization (BN) layers to improve the compute-to-communication (CTC) ratio and reduce the energy cost of off-chip memory access (MA). However, the training of activation layers remains challenging due to the additional off-chip MA required for derivative calculations. This article proposes MASL-AFU, an architecture designed to accelerate the activation layer in on-device DNN training. MASL-AFU leverages nonuniform piecewise linear (NUPWL) functions to speed up the forward propagation (FP) in the activation layer. During the error propagation (EP) process, retrieving derivatives from a lookup table (LUT) eliminates the need for redundant retrieval of the input data used in FP. By storing LUT indices instead of the original activation inputs, MASL-AFU significantly reduces and accelerates MA. Compared to other activation function units, MASL-AFU offers up to a 5.8x increase in computational and off-chip MA efficiency. In addition, MASL-AFU incorporates two dimensions of scalability: data precision and the number of LUT entries. These scalable, hardware-friendly methods enhance MASL-AFU's area efficiency by up to 3.24x and energy efficiency by up to 3.85x.

关键词： Training Artificial neural networks Table lookup Convolution Scalability Memory management Energy consumption Artificial intelligence Optimization Computational modeling Activation units deep neural network (DNN) training accelerator lookup table (LUT) on-device piecewise linear function

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Fuzzy portfolio selection in the trade-off between different economy trends and security returns

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APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 2025年第1期33卷

作者： Huang, Yin-Yin Pan, Jingchao Tsaur, Ruey-Chyn Lin, Nei-Chih Nanchang Vocat Univ Sch Econ & Management Nanchang Peoples R China Gongqing Inst Sci & Technol Gongqing City Peoples R China Tamkang Univ Dept Management Sci New Taipei 25137 Taiwan

Portfolio selection has been applied to the allocation of assets among the selected securities for obtaining the most profitable return. However, the theory of behavioural finance proposes that the investment proportions of securities are not only determined by the intrinsic value of the security but also largely affected by the behaviour of investors. Therefore, the behaviour analysis of an investor should play a major role in the portfolio selection. In this study, we apply a piecewise linear function to decompose the fuzzy expected return rates by judging the economy trend which transits from the current stage into the next stage and an upward or downward trend for each security return in the next stage. Analysis results indicated that we could obtain the investment proportions for the selected securities in different transitions to the economy stage with the judgement of upward or downward trend to each security. The portfolio selection in which the economy transits to the recovery stage derived a higher expected return rate than the other stages. Finally, in the illustration, any investor can use the selected portfolio by his investment behaviour after collecting the required information for the economy stage and return trends of securities.

关键词： Fuzzy portfolio selection ehavioural finance piecewise linear function economy stage

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